Operations with radicals

Simplify the following mathematical expression by extracting factors from the radicals and grouping like terms:

$5\sqrt{12} - \sqrt{27} + 2\sqrt{75}$

A technology company is designing a new smartphone model. The screen is rectangular and its exact dimensions are given by the following radicals: the base measures $\sqrt{12}$ cm and the height measures $\sqrt{3}$ cm. Calculate the area of the screen, expressing the result in simplified form.

An interior decorator wants to place a wooden moulding around a large square mirror that has an area of $20 \text{ m}^2$. Express the perimeter of the mirror in simplified radical form (extract factors if necessary).

In an assessment test, the teacher asks to simplify the following sum of radicals:

$$\sqrt{18} + \sqrt{50} - \sqrt{8}$$

Two students submit incorrect answers:

• Seán writes: “Since the radicands are different, I apply the property of the square root of a sum and get $\sqrt{18 + 50 - 8} = \sqrt{60} = 2\sqrt{15}$”.

• Aoife states: “They are not like radicals because they have different radicands ($18 \neq 50 \neq 8$), so the expression cannot be grouped or simplified any further; it must be left with the three terms”.

1. a) Explain precisely the algebraic error made by Seán regarding the root of a sum.
2. b) Explain Aoife's conceptual error regarding the criterion needed to determine whether two or more radicals are like radicals.
3. c) Solve the expression correctly: factor out all possible factors from each root, justify how the distributive property is applied to group them, and calculate the final simplified result.

Perform the following operations with radicals of the same index and express the result in the simplest possible form:

1. a) $\sqrt[4]{2} \times \sqrt[4]{8}$

1. b) $\frac{\sqrt[3]{81}}{\sqrt[3]{3}}$

Extract all possible factors from the following radicals to obtain the simplest expression:

1. a) $\sqrt{72x^4y^3}$

1. b) $\sqrt[3]{128a^5}$

Apply the properties of radicals to reduce to a single root and simplify the final result:

$\sqrt[3]{\sqrt{64x^{12}}}$

Insert the factors inside the radical in the following expressions:

1. a) $3 \sqrt{5x}$

1. b) $2a^2 \sqrt[3]{3a}$

Extract all possible factors from the following radicals and simplify the resulting expression to obtain a single term:

$3\sqrt{80} - 2\sqrt{45} + \sqrt{20}$

In the design of a new smartphone chip, one of the connections has a length given by the product of two radical components. Calculate the exact value of this length by simplifying the following product and expressing it under a single radical:

$\sqrt[3]{a^2} \times \sqrt[4]{a^3}$

Reduce the following expression to a single radical as simple as possible, applying the properties of a root of a root and the extraction of factors:

$\sqrt{\sqrt[3]{64x^8}}$

We want to simplify the following combined operation with radicals of different indices to express it as a single irreducible radical:

$$P = \frac{\sqrt[3]{4} \times \sqrt{8}}{\sqrt[6]{32}}$$

Two classmates discuss the most suitable strategy:

• Method A (Reduction to a common index): Factor the radicands into prime factors, calculate the least common multiple of the indices, and unify everything under a single root before simplifying inside the radicand.

• Method B (Fractional exponents): Express each root as a power with base 2 and a fractional exponent, and apply the properties of the product and quotient of powers with the same base by adding and subtracting exponents.

1. a) Solve the expression step by step using Method A.
2. b) Solve the expression step by step using Method B.
3. c) Justify with reasoned arguments which of the two methods you consider more efficient and with less risk of calculation error in this specific case.
4. d) Would Method B still be as directly applicable if, instead of the number 8, you had the number 12? Justify what change would occur in the procedure with both methods.

Rationalise the following fractions so that no radical remains in the denominator. Remember to simplify the final result if possible:

1. a) $\frac{10}{\sqrt{2}}$

1. b) $\frac{3}{\sqrt[3]{5}}$

Rationalise the following expressions to eliminate the radicals from the denominator and simplify the final result:

1. a) $\dfrac{6}{\sqrt[3]{9}}$

1. b) $\dfrac{10}{\sqrt{5}}$

Consider the following numerical expression with radicals:

$$E = \frac{10}{\sqrt{5}} - \sqrt{45} + \frac{4}{\sqrt{5}-1}$$

1. a) Estimate: Without performing exact algebraic transformations and knowing that $2.2 < \sqrt{5} < 2.3$ (you can take $\sqrt{5} \approx 2.2$):

• Estimate the value of the first term by dividing $10$ by $2.2$.

• Estimate the value of the second term $\sqrt{45}$ knowing that it lies between $\sqrt{36} = 6$ and $\sqrt{49} = 7$.

• Estimate the value of the third term by dividing $4$ by $(\sqrt{5}-1) \approx 1.2$.

• From the three estimates, predict between which two integers the overall result of $E$ will lie.

1. b) Check and calculate: Rationalize the first term (monomial denominator), factor out terms from the second term, and rationalize the third term by multiplying by its conjugate. Group like radicals and find the simplified exact value.
2. c) Compare the exact value obtained with your previous estimate and assess the accuracy of your initial reasoning.

Calculate the area of a square public square whose side measures $(2 + \sqrt{5})$ metres. Express the result in exact form, simplifying any radicals that appear when expanding the binomial.

An architect needs to calculate the pressure exerted by a structure on the ground. After completing the initial calculations, the resulting expression is as follows. Rationalise it using the conjugate of the denominator and simplify it as much as possible:

$\dfrac{4}{\sqrt{7} - \sqrt{3}}$

Perform the following combined operation with radicals, applying the distributive property and simplifying like terms:

$(\sqrt{3} + 2) \times (2\sqrt{3} - 5)$

Rationalise and simplify the following fraction. You must work out the numerator after rationalising to obtain the simplest possible expression:

$$\frac{2\sqrt{6} - 3\sqrt{2}}{2\sqrt{3} + \sqrt{2}}$$

Rationalise the following expression. Note that, as there are three square roots in the denominator, you may need to multiply by the conjugate more than once:

$$\frac{4}{\sqrt{2} + \sqrt{3} - \sqrt{5}}$$

Simplify the following combined expression as much as possible, extracting factors from the radicals and rationalising any terms necessary to group the numerical part:

$$3\sqrt{175} - \frac{5}{\sqrt{7}} + \sqrt[4]{441} - 2\sqrt{112}$$

Reduce to a single radical and simplify the resulting expression, assuming that the variables $x$ and $y$ represent positive real values:

$$\frac{\sqrt[3]{x^2 \times y} \times \sqrt{x \times y^3}}{\sqrt[6]{x^5 \times y^4}}$$

An architect is designing a triangular skylight for a new shopping centre. The design requires the base of the skylight to be an equilateral triangle whose height is exactly $H = \sqrt{18} + \sqrt{6}$ metres. Calculate the exact value of the area of the skylight, expressing the result as a simplified radical in $m^2$.